cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A036691 Compositorial numbers: product of first n composite numbers.

Original entry on oeis.org

1, 4, 24, 192, 1728, 17280, 207360, 2903040, 43545600, 696729600, 12541132800, 250822656000, 5267275776000, 115880067072000, 2781121609728000, 69528040243200000, 1807729046323200000, 48808684250726400000, 1366643159020339200000
Offset: 0

Views

Author

Felice Russo

Keywords

Comments

a(A196415(n)) = A141092(n) * A053767(A196415(n)). - Reinhard Zumkeller, Oct 03 2011
For n>11, A000142(n) < a(n) < A002110(n). - Chayim Lowen, Aug 18 2015
For n = {2,3,4}, a(n) is testably a Zumkeller number (A083207). For n > 4, a(n) is of the form 2^e_1 * p_2^e_2 * … * p_m^e_m, where e_m = 1 and e = floor(log_2(p_m)) < e_1. Therefore, 2^e * p_m^e_m is primitive Zumkeler number (A180332). Therefore, 2^e_1 * p_m^e_m is a Zumkeller number. Therefore, a(n) = 2^e_1 * p_m^e_m * r, where r is relatively prime to 2*p_m is a Zumkeller number. Therefore, for n > 1, a(n) is a Zumkeller number (see my proof at A002182 for details). - Ivan N. Ianakiev, May 04 2020

Examples

			a(3) = c(1)*c(2)*c(3) = 4*6*8 = 192.

Links

Crossrefs

Cf. primorial numbers A002110. Distinct members of A049614. See also A049650, A060880.
Cf. A092435 (subsequence: A092435(n) = a(prime(n)-n-1)). - Chayim Lowen, Jul 23 2015

Programs

  • Haskell
    a036691_list = scanl1 (*) a002808_list -- Reinhard Zumkeller, Oct 03 2011
  • Maple
    A036691 := proc(n)
        mul(A002808(i),i=1..n) ;
end proc: # R. J. Mathar, Oct 03 2011
  • Mathematica
    Composite[n_] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; Table[ Product[ Composite[i], {i, 1, n}], {n, 0, 18}] (* Robert G. Wilson v, Sep 13 2003 *)
nn=50;cnos=Complement[Range[nn],Prime[Range[PrimePi[nn]]]];Rest[FoldList[ Times,1,cnos]] (* Harvey P. Dale, May 19 2011 *)
A036691 = Union[Table[n!/(Times@@Prime[Range[PrimePi[n]]]), {n, 29}]] (* Alonso del Arte, Sep 21 2011 *)
Join[{1},FoldList[Times,Select[Range[30],CompositeQ]]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 14 2019 *)
  • PARI
    a(n)=my(c,p);c=4;p=1;while(n>0,if(!isprime(c),p=p*c;n=n-1);c=c+1);p \\ Ralf Stephan, Dec 21 2013
  • Python
    from sympy import factorial, primepi, primorial, composite
def A036691(n):
    return factorial(composite(n))//primorial(primepi(composite(n))) if n > 0 else 1 # Chai Wah Wu, Sep 08 2020

Formula

From Chayim Lowen, Jul 23 - Aug 05 2015: (Start)
a(n) = A049614(A002808(n)) = A000142(A002808(n))/A034386(A002808(n)).
a(n) = Product_{k=1..A002808(n)-n-1} prime(k)^(A085604(A002808(n),k)-1).
Sum_{k >= 1} 1/a(k) = 1.2975167655550616507663335821769... is to this sequence as e is to the factorials. (End)

Extensions

Corrected and extended by Niklas Eriksen (f95-ner(AT)nada.kth.se) and N. J. A. Sloane

A103855 a(n) = prime(n)! - prime(n)# + 1.

Original entry on oeis.org

1, 1, 91, 4831, 39914491, 6226990771, 355687427585491, 121645100399132311, 25852016738884753547131, 8841761993739701954537146306771, 8222838654177922817725362319509871, 13763753091226345046315979581573481661865191, 33452526613163807108170062053440751360901736472791
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 20 2005

Keywords

Links

Crossrefs

Cf. A103856, A103857, A103858, A103859, A103860, A103861, A103890.
Cf. A002110, A039716, A092435.

Programs

  • Mathematica
    primorial[n_] := Product[Prime[i], {i, n}]; A103855[n_] := Prime[n]! - primorial[n] + 1; Array[A103855, 20] (* G. C. Greubel, May 09 2017 *)
With[{nn=15},#[[1]]-#[[2]]+1&/@Thread[{Prime[Range[nn]]!,FoldList[Times,Prime[Range[nn]]]}]] (* Harvey P. Dale, Aug 11 2025 *)

Formula

a(n) = A039716(n) - A002110(n) + 1 = A002110(n) * (A092435(n) - 1) + 1.

A103890 a(n) = prime(n)! / prime(n)# + 1.

Original entry on oeis.org

2, 2, 5, 25, 17281, 207361, 696729601, 12541132801, 115880067072001, 1366643159020339200001, 40999294770610176000001, 1854768736099424576471040000001, 109950690675973888893203251200000001, 4617929008390903333514536550400000001
Offset: 1

Views

Author

Reinhard Zumkeller, Feb 20 2005

Keywords

Links

Crossrefs

Cf. A103855, A103891, A103892, A103893, A103894, A103895, A103896.

Programs

  • Mathematica
    #[[1]]/#[[2]]&/@With[{nn=15},Thread[{Prime[Range[nn]]!,FoldList[ Times,Prime[ Range[nn]]]}]]+1 (* Harvey P. Dale, May 21 2019 *)
  • PARI
    a(n) = prime(n)!/vecprod(primes(n)) + 1; \\ Michel Marcus, Nov 12 2023

Formula

a(n) = A039716(n)/A002110(n) + 1 = A092435(n) + 1.

A109915 Product of all composite numbers k such that n

Original entry on oeis.org

1, 1, 4, 1, 6, 1, 720, 90, 10, 1, 12, 1, 3360, 240, 16, 1, 18, 1, 9240, 462, 22, 1, 11793600, 491400, 19656, 756, 28, 1, 30, 1, 45239040, 1413720, 42840, 1260, 36, 1, 59280, 1560, 40, 1, 42, 1, 91080, 2070, 46, 1, 311875200, 6497400, 132600, 2652
Offset: 1

Views

Author

Amarnath Murthy, Jul 16 2005

Keywords

Comments

All terms that differ from 1 have the format A092435(i+1)/A092435(i). - R. J. Mathar, Aug 15 2007

Examples

			a(7) = 8*9*10 = 720.

Links

Crossrefs

Cf. A109914.

Programs

  • Maple
    A109915 := proc(n) local a,rm1,k; a := 1: rm1 := numtheory[pi](n) ; for k from n+1 to ithprime(rm1+1) do if not isprime(k) then a := a*k; fi; od: RETURN(a) ; end: seq(A109915(n),n=1..50) ; # R. J. Mathar, Aug 15 2007

Extensions

Corrected and extended by R. J. Mathar, Aug 15 2007

A124082 Numbers k such that prime(k)!/prime(k)# - 1 is prime.

Original entry on oeis.org

3, 4, 7, 21, 60
Offset: 1

Views

Author

Pierre CAMI, Nov 25 2006

Keywords

Comments

No more terms through 1000. - Ryan Propper, Jan 27 2007
No more terms through 2500. - Michael S. Branicky, Oct 02 2024

Examples

			1*2*3*4*5/(2*3*5) - 1 = 3, a prime, so a(1)=3 as 5=prime(3);
1*2*3*4*5*6*7/(2*3*5*7) - 1 = 23, a prime, so a(2)=4 as 7=prime(4);
1*2*3*4*5*6*7*8*9*10*11*12*13*14*15*16*17/(2*3*5*7*11*13*17) - 1 = 696729599, a prime, so a(3)=7 as 17=prime(7).

Crossrefs

Cf. A092435 (p!/p#), A124083.

Programs

  • Mathematica
    Primorial[n_] := Times @@ Select[Range[n], PrimeQ]; Do[k = Prime[n]; If[PrimeQ[k!/Primorial[k] - 1], Print[n]], {n, 10^3}] (* Ryan Propper, Jan 27 2007 *)
With[{nn=70},Position[#[[1]]/#[[2]]-1&/@Thread[{Prime[Range[ nn]]!,FoldList[ Times,Prime[Range[nn]]]}],?PrimeQ]//Flatten] (* _Harvey P. Dale, Jul 01 2020 *)
  • PARI
    isok(k) = isprime(prime(k)!/prod(i=1, k, prime(i)) - 1); \\ Michel Marcus, Sep 15 2019

A124083 Numbers k such that prime(k)!/prime(k)# + 1 is prime.

Original entry on oeis.org

1, 2, 3, 60, 90
Offset: 1

Views

Author

Pierre CAMI, Nov 25 2006

Keywords

Comments

a(6) > 500 if it exists. - Felix Fröhlich, Sep 15 2019
a(6) > 2500 if it exists. - Michael S. Branicky, Oct 01 2024

Examples

			1*2/2 +1 = 2 prime so a(1)=1 as 2=prime(1).
1*2*3/(2*3) +1 = 2 prime so a(2)=2 as 3=prime(2).
1*2*3*4*5/(2*3*5) +1 = 5 prime so a(3)=3 as 5=prime(3).

Crossrefs

Cf. A092435 (p!/p#), A124082.

Programs

  • PARI
    isok(k) = isprime(prime(k)!/prod(i=1, k, prime(i)) + 1); \\ Michel Marcus, Sep 15 2019

A249241 a(n) = p - prime(n)!/prime(n)#, where p is the smallest prime number > prime(n)!/prime(n)#+1.

Original entry on oeis.org

2, 2, 3, 5, 11, 7, 29, 17, 17, 397, 47, 67, 23, 41, 31, 157, 409, 31, 151, 109, 199, 191, 131, 61, 103, 547, 179, 269, 389, 317, 181, 331, 307, 173, 1259, 1289, 619, 131, 223, 683, 139, 241, 191, 101, 1039, 1367, 1153, 241, 1187, 479, 149, 181, 487, 1093, 571, 1151, 809, 199, 823, 491, 191, 151, 1321, 197, 163, 337, 467, 659, 673, 877, 487, 743, 313, 673, 857, 677, 1021
Offset: 1

Views

Author

Werner D. Sand, Oct 23 2014

Keywords

Comments

Conjecture: All terms are prime.
While Fortune's conjecture (A005235) uses products of primes, this sequence uses products of composite numbers (more exactly: of nonprimes, because 1 belongs to them). It looks like all multiples of prime(n)# (except some powers) lead to a sequence which contains only prime numbers.

Examples

			n = 1; prime(1)!/prime(1)# = 2/2 = 1; p = nextprime(1+1) = 3; a(1) = 3-1 = 2.

Crossrefs

Cf. A092435.

Programs

  • MuPAD
    q:=1; p:=1; for i from 1 to 100 do q:=nextprime(q+1); p:=p*q; N:=nextprime((fact(q)/p)+2)-fact(q)/p; print(i,N); end_for:
  • PARI
    A092435(n)=prime(n)!/prod(i=1,n,prime(i))
a(n)=my(t=A092435(n)); nextprime(t+2)-t \\ Charles R Greathouse IV, Oct 23 2014
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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